Commentary on Proposition X.47

Back to Proposition X.47.

[Folio 0105 | p. 101]

Using the same notation as in the note on X. 44, we suppose that, if possible,

$$x + y = x' + y',$$

and we put

$$\left. \begin{array}{r} x^2 + y^2 = \sigma u \\ 2xy = \sigma v \end{array} \right\} \quad \text{and} \quad \left. \begin{array}{r} x'^2 + y'^2 = \sigma u' \\ 2x'y' = \sigma v' \end{array} \right\}.$$

Then, since $x^2 + y^2, 2xy$ are medial areas, and $\sigma$ rational, $u, v$ are both rational and $\smile \sigma$ ………………………………….(1).

Also, by hypothesis,

$$x^2 + y^2 \smile 2xy,$$

whence

$$u \smile v \text{ .................................................................(2).}$$

Therefore, by (1) and (2), $u, v$ are rational and $\sim$. Hence $u + v$ is a binomial straight line. [X. 36] Similarly $u' + v'$ is a binomial straight line. But

$$u + v = u' + v';$$

therefore a binomial straight line is divided into terms in two ways: which is impossible. [X. 42]

Therefore etc.